

A212629


Number of vertices in all maximal independent vertex subsets in the rooted tree with MatulaGoebel number n.


11



1, 2, 3, 3, 6, 6, 4, 4, 9, 9, 9, 8, 8, 8, 14, 5, 8, 15, 5, 11, 11, 14, 15, 10, 22, 15, 25, 10, 11, 20, 14, 6, 22, 11, 17, 19, 10, 10, 20, 13, 15, 18, 10, 17, 33, 25, 20, 12, 13, 28, 17, 18, 6, 36, 32, 12, 13, 20, 11, 24, 19, 22, 29, 7, 31, 31, 10, 13, 33, 24, 13, 23, 18, 19, 45, 12, 26, 32
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OFFSET

1,2


COMMENTS

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. An independent vertex subset S of a tree is said to be maximal if every vertex that is not in S is joined by an edge to at least one vertex of S.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the MatulaGoebel numbers of the m branches of T.
Let A(n)=A(n,x), B(n)=B(n,x), C(n)=C(n,x) be the generating polynomial with respect to size of the maximal independent sets that contain the root, the maximal independent sets that do not contain the root, and the independent sets which are not maximal but become maximal if the root is removed, respectively. We have: A(1)=x, B(1)=0, C(1)=1, A(tth prime) = x[B(t) + C(t)], B(tth prime) = A(t), C(tth prime)=B(t), A(rs)=A(r)A(s)/x, B(rs)=B(r)B(s)+B(r)C(s)+B(s)C(r), C(rs)=C(r)C(s) (r,s>=2). The generating polynomial of the maximal independent vertex subsets with respect to size is P(n, x)=A(n,x)+B(n,x). Then a(n)=subs(x=1, dP(n,x)/dx). The Maple program is based on these relations.


LINKS

Table of n, a(n) for n=1..78.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
H. S. Wilf, The number of maximal independent sets in a tree, SIAM J. Alg. Disc. Math., 7, 1986, 125130.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(n) = Sum(k*A212627(n,k), k>=1).


EXAMPLE

a(11)=9 because the rooted tree with MatulaGoebel number 11 is the path tree on 5 vertices R  A  B  C  D; the maximal independent vertex subsets are {R,C}, {A,C}, {A,D}, and {R,B,D}.


MAPLE

with(numtheory): P := proc (n) local r, s, A, B, C: r := n> op(1, factorset(n)): s := n> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: if n = 1 then x else sort(expand(A(n)+B(n))) end if end proc: seq(subs(x = 1, diff(P(n), x)), n = 1 .. 120);


CROSSREFS

Cf. A206499, A212627, A212618A212632.
Sequence in context: A347732 A075258 A321745 * A127779 A207634 A222862
Adjacent sequences: A212626 A212627 A212628 * A212630 A212631 A212632


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 08 2012


EXTENSIONS

Sequences A212618A212632 edited by M. Marcus and M. F. Hasler, Jan 06 2013


STATUS

approved



